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Asymptotic variance of Newton–Cotes quadratures based on randomized sampling points

Part of: Special processes Stochastic processes Numerical approximation and computational geometry

Published online by Cambridge University Press:  03 December 2020

Mads Stehr  and
Markus Kiderlen
Mads Stehr*
Affiliation:
Aarhus University
Markus Kiderlen*
Affiliation:
Aarhus University
*
*Postal address: Centre for Stochastic Geometry and Advanced Bioimaging (CSGB), Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark.
*Postal address: Centre for Stochastic Geometry and Advanced Bioimaging (CSGB), Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark.
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Abstract

We consider the problem of numerical integration when the sampling nodes form a stationary point process on the real line. In previous papers it was argued that a naïve Riemann sum approach can cause a severe variance inflation when the sampling points are not equidistant. We show that this inflation can be avoided using a higher-order Newton–Cotes quadrature rule which exploits smoothness properties of the integrand. Under mild assumptions, the resulting estimator is unbiased and its variance asymptotically obeys a power law as a function of the mean point distance. If the Newton–Cotes rule is of sufficiently high order, the exponent of this law turns out to only depend on the point process through its mean point distance. We illustrate our findings with the stereological estimation of the volume of a compact object, suggesting alternatives to the well-established Cavalieri estimator.

Keywords

Point processstationary stochastic processrandomized Newton–Cotes quadraturenumerical integrationasymptotic variance boundsrenewal process

MSC classification

Primary: 65D30: Numerical integration 60G55: Point processes
Secondary: 60G10: Stationary processes 60K05: Renewal theory 65D32: Quadrature and cubature formulas
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 4 , December 2020 , pp. 1284 - 1307
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Copyright
© Applied Probability Trust 2020

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